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                        <h1 class="title">Data Types - MLlib</h1>
                    

                    <ul id="markdown-toc">
  <li><a href="#local-vector">Local vector</a></li>
  <li><a href="#labeled-point">Labeled point</a></li>
  <li><a href="#local-matrix">Local matrix</a></li>
  <li><a href="#distributed-matrix">Distributed matrix</a>    <ul>
      <li><a href="#rowmatrix">RowMatrix</a></li>
      <li><a href="#indexedrowmatrix">IndexedRowMatrix</a></li>
      <li><a href="#coordinatematrix">CoordinateMatrix</a></li>
      <li><a href="#blockmatrix">BlockMatrix</a></li>
    </ul>
  </li>
</ul>

<p>MLlib supports local vectors and matrices stored on a single machine, 
as well as distributed matrices backed by one or more RDDs.
Local vectors and local matrices are simple data models 
that serve as public interfaces. The underlying linear algebra operations are provided by
<a href="http://www.scalanlp.org/">Breeze</a> and <a href="http://jblas.org/">jblas</a>.
A training example used in supervised learning is called a &#8220;labeled point&#8221; in MLlib.</p>

<h2 id="local-vector">Local vector</h2>

<p>A local vector has integer-typed and 0-based indices and double-typed values, stored on a single
machine.  MLlib supports two types of local vectors: dense and sparse.  A dense vector is backed by
a double array representing its entry values, while a sparse vector is backed by two parallel
arrays: indices and values.  For example, a vector <code>(1.0, 0.0, 3.0)</code> can be represented in dense
format as <code>[1.0, 0.0, 3.0]</code> or in sparse format as <code>(3, [0, 2], [1.0, 3.0])</code>, where <code>3</code> is the size
of the vector.</p>

<div class="codetabs">
<div data-lang="scala">

    <p>The base class of local vectors is
<a href="api/scala/index.html#org.apache.spark.mllib.linalg.Vector"><code>Vector</code></a>, and we provide two
implementations: <a href="api/scala/index.html#org.apache.spark.mllib.linalg.DenseVector"><code>DenseVector</code></a> and
<a href="api/scala/index.html#org.apache.spark.mllib.linalg.SparseVector"><code>SparseVector</code></a>.  We recommend
using the factory methods implemented in
<a href="api/scala/index.html#org.apache.spark.mllib.linalg.Vectors$"><code>Vectors</code></a> to create local vectors.</p>

    <p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.linalg.Vector"><code>Vector</code> Scala docs</a> and <a href="api/scala/index.html#org.apache.spark.mllib.linalg.Vectors"><code>Vectors</code> Scala docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.</span><span class="o">{</span><span class="nc">Vector</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">}</span>

<span class="c1">// Create a dense vector (1.0, 0.0, 3.0).</span>
<span class="k">val</span> <span class="n">dv</span><span class="k">:</span> <span class="kt">Vector</span> <span class="o">=</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">0.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">)</span>
<span class="c1">// Create a sparse vector (1.0, 0.0, 3.0) by specifying its indices and values corresponding to nonzero entries.</span>
<span class="k">val</span> <span class="n">sv1</span><span class="k">:</span> <span class="kt">Vector</span> <span class="o">=</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">sparse</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="nc">Array</span><span class="o">(</span><span class="mi">0</span><span class="o">,</span> <span class="mi">2</span><span class="o">),</span> <span class="nc">Array</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">))</span>
<span class="c1">// Create a sparse vector (1.0, 0.0, 3.0) by specifying its nonzero entries.</span>
<span class="k">val</span> <span class="n">sv2</span><span class="k">:</span> <span class="kt">Vector</span> <span class="o">=</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">sparse</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="nc">Seq</span><span class="o">((</span><span class="mi">0</span><span class="o">,</span> <span class="mf">1.0</span><span class="o">),</span> <span class="o">(</span><span class="mi">2</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">)))</span></code></pre></div>

    <p><strong><em>Note:</em></strong>
Scala imports <code>scala.collection.immutable.Vector</code> by default, so you have to import
<code>org.apache.spark.mllib.linalg.Vector</code> explicitly to use MLlib&#8217;s <code>Vector</code>.</p>

  </div>

<div data-lang="java">

    <p>The base class of local vectors is
<a href="api/java/org/apache/spark/mllib/linalg/Vector.html"><code>Vector</code></a>, and we provide two
implementations: <a href="api/java/org/apache/spark/mllib/linalg/DenseVector.html"><code>DenseVector</code></a> and
<a href="api/java/org/apache/spark/mllib/linalg/SparseVector.html"><code>SparseVector</code></a>.  We recommend
using the factory methods implemented in
<a href="api/java/org/apache/spark/mllib/linalg/Vectors.html"><code>Vectors</code></a> to create local vectors.</p>

    <p>Refer to the <a href="api/java/org/apache/spark/mllib/linalg/Vector.html"><code>Vector</code> Java docs</a> and <a href="api/java/org/apache/spark/mllib/linalg/Vectors.html"><code>Vectors</code> Java docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span><span class="o">;</span>

<span class="c1">// Create a dense vector (1.0, 0.0, 3.0).</span>
<span class="n">Vector</span> <span class="n">dv</span> <span class="o">=</span> <span class="n">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">0.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">);</span>
<span class="c1">// Create a sparse vector (1.0, 0.0, 3.0) by specifying its indices and values corresponding to nonzero entries.</span>
<span class="n">Vector</span> <span class="n">sv</span> <span class="o">=</span> <span class="n">Vectors</span><span class="o">.</span><span class="na">sparse</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="k">new</span> <span class="kt">int</span><span class="o">[]</span> <span class="o">{</span><span class="mi">0</span><span class="o">,</span> <span class="mi">2</span><span class="o">},</span> <span class="k">new</span> <span class="kt">double</span><span class="o">[]</span> <span class="o">{</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">});</span></code></pre></div>

  </div>

<div data-lang="python">
    <p>MLlib recognizes the following types as dense vectors:</p>

    <ul>
      <li>NumPy&#8217;s <a href="http://docs.scipy.org/doc/numpy/reference/generated/numpy.array.html"><code>array</code></a></li>
      <li>Python&#8217;s list, e.g., <code>[1, 2, 3]</code></li>
    </ul>

    <p>and the following as sparse vectors:</p>

    <ul>
      <li>MLlib&#8217;s <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.SparseVector"><code>SparseVector</code></a>.</li>
      <li>SciPy&#8217;s
<a href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.csc_matrix.html#scipy.sparse.csc_matrix"><code>csc_matrix</code></a>
with a single column</li>
    </ul>

    <p>We recommend using NumPy arrays over lists for efficiency, and using the factory methods implemented
in <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.Vectors"><code>Vectors</code></a> to create sparse vectors.</p>

    <p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.Vectors"><code>Vectors</code> Python docs</a> for more details on the API.</p>

    <div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">scipy.sparse</span> <span class="kn">as</span> <span class="nn">sps</span>
<span class="kn">from</span> <span class="nn">pyspark.mllib.linalg</span> <span class="kn">import</span> <span class="n">Vectors</span>

<span class="c"># Use a NumPy array as a dense vector.</span>
<span class="n">dv1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">])</span>
<span class="c"># Use a Python list as a dense vector.</span>
<span class="n">dv2</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">]</span>
<span class="c"># Create a SparseVector.</span>
<span class="n">sv1</span> <span class="o">=</span> <span class="n">Vectors</span><span class="o">.</span><span class="n">sparse</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">])</span>
<span class="c"># Use a single-column SciPy csc_matrix as a sparse vector.</span>
<span class="n">sv2</span> <span class="o">=</span> <span class="n">sps</span><span class="o">.</span><span class="n">csc_matrix</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">]),</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]),</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])),</span> <span class="n">shape</span> <span class="o">=</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span></code></pre></div>

  </div>
</div>

<h2 id="labeled-point">Labeled point</h2>

<p>A labeled point is a local vector, either dense or sparse, associated with a label/response.
In MLlib, labeled points are used in supervised learning algorithms.
We use a double to store a label, so we can use labeled points in both regression and classification.
For binary classification, a label should be either <code>0</code> (negative) or <code>1</code> (positive).
For multiclass classification, labels should be class indices starting from zero: <code>0, 1, 2, ...</code>.</p>

<div class="codetabs">

<div data-lang="scala">

    <p>A labeled point is represented by the case class
<a href="api/scala/index.html#org.apache.spark.mllib.regression.LabeledPoint"><code>LabeledPoint</code></a>.</p>

    <p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.regression.LabeledPoint"><code>LabeledPoint</code> Scala docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span>

<span class="c1">// Create a labeled point with a positive label and a dense feature vector.</span>
<span class="k">val</span> <span class="n">pos</span> <span class="k">=</span> <span class="nc">LabeledPoint</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">0.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">))</span>

<span class="c1">// Create a labeled point with a negative label and a sparse feature vector.</span>
<span class="k">val</span> <span class="n">neg</span> <span class="k">=</span> <span class="nc">LabeledPoint</span><span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">sparse</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="nc">Array</span><span class="o">(</span><span class="mi">0</span><span class="o">,</span> <span class="mi">2</span><span class="o">),</span> <span class="nc">Array</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">)))</span></code></pre></div>

  </div>

<div data-lang="java">

    <p>A labeled point is represented by
<a href="api/java/org/apache/spark/mllib/regression/LabeledPoint.html"><code>LabeledPoint</code></a>.</p>

    <p>Refer to the <a href="api/java/org/apache/spark/mllib/regression/LabeledPoint.html"><code>LabeledPoint</code> Java docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span><span class="o">;</span>

<span class="c1">// Create a labeled point with a positive label and a dense feature vector.</span>
<span class="n">LabeledPoint</span> <span class="n">pos</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">LabeledPoint</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="n">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">0.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">));</span>

<span class="c1">// Create a labeled point with a negative label and a sparse feature vector.</span>
<span class="n">LabeledPoint</span> <span class="n">neg</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">LabeledPoint</span><span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="n">Vectors</span><span class="o">.</span><span class="na">sparse</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="k">new</span> <span class="kt">int</span><span class="o">[]</span> <span class="o">{</span><span class="mi">0</span><span class="o">,</span> <span class="mi">2</span><span class="o">},</span> <span class="k">new</span> <span class="kt">double</span><span class="o">[]</span> <span class="o">{</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">}));</span></code></pre></div>

  </div>

<div data-lang="python">

    <p>A labeled point is represented by
<a href="api/python/pyspark.mllib.html#pyspark.mllib.regression.LabeledPoint"><code>LabeledPoint</code></a>.</p>

    <p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.regression.LabeledPoint"><code>LabeledPoint</code> Python docs</a> for more details on the API.</p>

    <div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.linalg</span> <span class="kn">import</span> <span class="n">SparseVector</span>
<span class="kn">from</span> <span class="nn">pyspark.mllib.regression</span> <span class="kn">import</span> <span class="n">LabeledPoint</span>

<span class="c"># Create a labeled point with a positive label and a dense feature vector.</span>
<span class="n">pos</span> <span class="o">=</span> <span class="n">LabeledPoint</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">])</span>

<span class="c"># Create a labeled point with a negative label and a sparse feature vector.</span>
<span class="n">neg</span> <span class="o">=</span> <span class="n">LabeledPoint</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">SparseVector</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">]))</span></code></pre></div>

  </div>
</div>

<p><strong><em>Sparse data</em></strong></p>

<p>It is very common in practice to have sparse training data.  MLlib supports reading training
examples stored in <code>LIBSVM</code> format, which is the default format used by
<a href="http://www.csie.ntu.edu.tw/~cjlin/libsvm/"><code>LIBSVM</code></a> and
<a href="http://www.csie.ntu.edu.tw/~cjlin/liblinear/"><code>LIBLINEAR</code></a>.  It is a text format in which each line
represents a labeled sparse feature vector using the following format:</p>

<pre><code>label index1:value1 index2:value2 ...
</code></pre>

<p>where the indices are one-based and in ascending order. 
After loading, the feature indices are converted to zero-based.</p>

<div class="codetabs">
<div data-lang="scala">

    <p><a href="api/scala/index.html#org.apache.spark.mllib.util.MLUtils$"><code>MLUtils.loadLibSVMFile</code></a> reads training
examples stored in LIBSVM format.</p>

    <p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.util.MLUtils"><code>MLUtils</code> Scala docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.util.MLUtils</span>
<span class="k">import</span> <span class="nn">org.apache.spark.rdd.RDD</span>

<span class="k">val</span> <span class="n">examples</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">LabeledPoint</span><span class="o">]</span> <span class="k">=</span> <span class="nc">MLUtils</span><span class="o">.</span><span class="n">loadLibSVMFile</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="s">&quot;data/mllib/sample_libsvm_data.txt&quot;</span><span class="o">)</span></code></pre></div>

  </div>

<div data-lang="java">
    <p><a href="api/java/org/apache/spark/mllib/util/MLUtils.html"><code>MLUtils.loadLibSVMFile</code></a> reads training
examples stored in LIBSVM format.</p>

    <p>Refer to the <a href="api/java/org/apache/spark/mllib/util/MLUtils.html"><code>MLUtils</code> Java docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.util.MLUtils</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaRDD</span><span class="o">;</span>

<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">LabeledPoint</span><span class="o">&gt;</span> <span class="n">examples</span> <span class="o">=</span> 
  <span class="n">MLUtils</span><span class="o">.</span><span class="na">loadLibSVMFile</span><span class="o">(</span><span class="n">jsc</span><span class="o">.</span><span class="na">sc</span><span class="o">(),</span> <span class="s">&quot;data/mllib/sample_libsvm_data.txt&quot;</span><span class="o">).</span><span class="na">toJavaRDD</span><span class="o">();</span></code></pre></div>

  </div>

<div data-lang="python">
    <p><a href="api/python/pyspark.mllib.html#pyspark.mllib.util.MLUtils"><code>MLUtils.loadLibSVMFile</code></a> reads training
examples stored in LIBSVM format.</p>

    <p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.util.MLUtils"><code>MLUtils</code> Python docs</a> for more details on the API.</p>

    <div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.util</span> <span class="kn">import</span> <span class="n">MLUtils</span>

<span class="n">examples</span> <span class="o">=</span> <span class="n">MLUtils</span><span class="o">.</span><span class="n">loadLibSVMFile</span><span class="p">(</span><span class="n">sc</span><span class="p">,</span> <span class="s">&quot;data/mllib/sample_libsvm_data.txt&quot;</span><span class="p">)</span></code></pre></div>

  </div>
</div>

<h2 id="local-matrix">Local matrix</h2>

<p>A local matrix has integer-typed row and column indices and double-typed values, stored on a single
machine.  MLlib supports dense matrices, whose entry values are stored in a single double array in
column-major order, and sparse matrices, whose non-zero entry values are stored in the Compressed Sparse
Column (CSC) format in column-major order.  For example, the following dense matrix <code>\[ \begin{pmatrix}
1.0 &amp; 2.0 \\
3.0 &amp; 4.0 \\
5.0 &amp; 6.0
\end{pmatrix}
\]</code>
is stored in a one-dimensional array <code>[1.0, 3.0, 5.0, 2.0, 4.0, 6.0]</code> with the matrix size <code>(3, 2)</code>.</p>

<div class="codetabs">
<div data-lang="scala">

    <p>The base class of local matrices is
<a href="api/scala/index.html#org.apache.spark.mllib.linalg.Matrix"><code>Matrix</code></a>, and we provide two
implementations: <a href="api/scala/index.html#org.apache.spark.mllib.linalg.DenseMatrix"><code>DenseMatrix</code></a>,
and <a href="api/scala/index.html#org.apache.spark.mllib.linalg.SparseMatrix"><code>SparseMatrix</code></a>.
We recommend using the factory methods implemented
in <a href="api/scala/index.html#org.apache.spark.mllib.linalg.Matrices$"><code>Matrices</code></a> to create local
matrices. Remember, local matrices in MLlib are stored in column-major order.</p>

    <p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.linalg.Matrix"><code>Matrix</code> Scala docs</a> and <a href="api/scala/index.html#org.apache.spark.mllib.linalg.Matrices"><code>Matrices</code> Scala docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.</span><span class="o">{</span><span class="nc">Matrix</span><span class="o">,</span> <span class="nc">Matrices</span><span class="o">}</span>

<span class="c1">// Create a dense matrix ((1.0, 2.0), (3.0, 4.0), (5.0, 6.0))</span>
<span class="k">val</span> <span class="n">dm</span><span class="k">:</span> <span class="kt">Matrix</span> <span class="o">=</span> <span class="nc">Matrices</span><span class="o">.</span><span class="n">dense</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="mi">2</span><span class="o">,</span> <span class="nc">Array</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">,</span> <span class="mf">5.0</span><span class="o">,</span> <span class="mf">2.0</span><span class="o">,</span> <span class="mf">4.0</span><span class="o">,</span> <span class="mf">6.0</span><span class="o">))</span>

<span class="c1">// Create a sparse matrix ((9.0, 0.0), (0.0, 8.0), (0.0, 6.0))</span>
<span class="k">val</span> <span class="n">sm</span><span class="k">:</span> <span class="kt">Matrix</span> <span class="o">=</span> <span class="nc">Matrices</span><span class="o">.</span><span class="n">sparse</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="mi">2</span><span class="o">,</span> <span class="nc">Array</span><span class="o">(</span><span class="mi">0</span><span class="o">,</span> <span class="mi">1</span><span class="o">,</span> <span class="mi">3</span><span class="o">),</span> <span class="nc">Array</span><span class="o">(</span><span class="mi">0</span><span class="o">,</span> <span class="mi">2</span><span class="o">,</span> <span class="mi">1</span><span class="o">),</span> <span class="nc">Array</span><span class="o">(</span><span class="mi">9</span><span class="o">,</span> <span class="mi">6</span><span class="o">,</span> <span class="mi">8</span><span class="o">))</span></code></pre></div>

  </div>

<div data-lang="java">

    <p>The base class of local matrices is
<a href="api/java/org/apache/spark/mllib/linalg/Matrix.html"><code>Matrix</code></a>, and we provide two
implementations: <a href="api/java/org/apache/spark/mllib/linalg/DenseMatrix.html"><code>DenseMatrix</code></a>,
and <a href="api/java/org/apache/spark/mllib/linalg/SparseMatrix.html"><code>SparseMatrix</code></a>.
We recommend using the factory methods implemented
in <a href="api/java/org/apache/spark/mllib/linalg/Matrices.html"><code>Matrices</code></a> to create local
matrices. Remember, local matrices in MLlib are stored in column-major order.</p>

    <p>Refer to the <a href="api/java/org/apache/spark/mllib/linalg/Matrix.html"><code>Matrix</code> Java docs</a> and <a href="api/java/org/apache/spark/mllib/linalg/Matrices.html"><code>Matrices</code> Java docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Matrix</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Matrices</span><span class="o">;</span>

<span class="c1">// Create a dense matrix ((1.0, 2.0), (3.0, 4.0), (5.0, 6.0))</span>
<span class="n">Matrix</span> <span class="n">dm</span> <span class="o">=</span> <span class="n">Matrices</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="mi">2</span><span class="o">,</span> <span class="k">new</span> <span class="kt">double</span><span class="o">[]</span> <span class="o">{</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">,</span> <span class="mf">5.0</span><span class="o">,</span> <span class="mf">2.0</span><span class="o">,</span> <span class="mf">4.0</span><span class="o">,</span> <span class="mf">6.0</span><span class="o">});</span>

<span class="c1">// Create a sparse matrix ((9.0, 0.0), (0.0, 8.0), (0.0, 6.0))</span>
<span class="n">Matrix</span> <span class="n">sm</span> <span class="o">=</span> <span class="n">Matrices</span><span class="o">.</span><span class="na">sparse</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="mi">2</span><span class="o">,</span> <span class="k">new</span> <span class="kt">int</span><span class="o">[]</span> <span class="o">{</span><span class="mi">0</span><span class="o">,</span> <span class="mi">1</span><span class="o">,</span> <span class="mi">3</span><span class="o">},</span> <span class="k">new</span> <span class="kt">int</span><span class="o">[]</span> <span class="o">{</span><span class="mi">0</span><span class="o">,</span> <span class="mi">2</span><span class="o">,</span> <span class="mi">1</span><span class="o">},</span> <span class="k">new</span> <span class="kt">double</span><span class="o">[]</span> <span class="o">{</span><span class="mi">9</span><span class="o">,</span> <span class="mi">6</span><span class="o">,</span> <span class="mi">8</span><span class="o">});</span></code></pre></div>

  </div>

<div data-lang="python">

    <p>The base class of local matrices is
<a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.Matrix"><code>Matrix</code></a>, and we provide two
implementations: <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.DenseMatrix"><code>DenseMatrix</code></a>,
and <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.SparseMatrix"><code>SparseMatrix</code></a>.
We recommend using the factory methods implemented
in <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.Matrices"><code>Matrices</code></a> to create local
matrices. Remember, local matrices in MLlib are stored in column-major order.</p>

    <p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.Matrix"><code>Matrix</code> Python docs</a> and <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.Matrices"><code>Matrices</code> Python docs</a> for more details on the API.</p>

    <div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.</span><span class="p">{</span><span class="n">Matrix</span><span class="p">,</span> <span class="n">Matrices</span><span class="p">}</span>

<span class="o">//</span> <span class="n">Create</span> <span class="n">a</span> <span class="n">dense</span> <span class="n">matrix</span> <span class="p">((</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">),</span> <span class="p">(</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">4.0</span><span class="p">),</span> <span class="p">(</span><span class="mf">5.0</span><span class="p">,</span> <span class="mf">6.0</span><span class="p">))</span>
<span class="n">dm2</span> <span class="o">=</span> <span class="n">Matrices</span><span class="o">.</span><span class="n">dense</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span>

<span class="o">//</span> <span class="n">Create</span> <span class="n">a</span> <span class="n">sparse</span> <span class="n">matrix</span> <span class="p">((</span><span class="mf">9.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">8.0</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">6.0</span><span class="p">))</span>
<span class="n">sm</span> <span class="o">=</span> <span class="n">Matrices</span><span class="o">.</span><span class="n">sparse</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">9</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">8</span><span class="p">])</span></code></pre></div>

  </div>

</div>

<h2 id="distributed-matrix">Distributed matrix</h2>

<p>A distributed matrix has long-typed row and column indices and double-typed values, stored
distributively in one or more RDDs.  It is very important to choose the right format to store large
and distributed matrices.  Converting a distributed matrix to a different format may require a
global shuffle, which is quite expensive.  Three types of distributed matrices have been implemented
so far.</p>

<p>The basic type is called <code>RowMatrix</code>. A <code>RowMatrix</code> is a row-oriented distributed
matrix without meaningful row indices, e.g., a collection of feature vectors.
It is backed by an RDD of its rows, where each row is a local vector.
We assume that the number of columns is not huge for a <code>RowMatrix</code> so that a single
local vector can be reasonably communicated to the driver and can also be stored /
operated on using a single node. 
An <code>IndexedRowMatrix</code> is similar to a <code>RowMatrix</code> but with row indices,
which can be used for identifying rows and executing joins.
A <code>CoordinateMatrix</code> is a distributed matrix stored in <a href="https://en.wikipedia.org/wiki/Sparse_matrix#Coordinate_list_.28COO.29">coordinate list (COO)</a> format,
backed by an RDD of its entries.</p>

<p><strong><em>Note</em></strong></p>

<p>The underlying RDDs of a distributed matrix must be deterministic, because we cache the matrix size.
In general the use of non-deterministic RDDs can lead to errors.</p>

<h3 id="rowmatrix">RowMatrix</h3>

<p>A <code>RowMatrix</code> is a row-oriented distributed matrix without meaningful row indices, backed by an RDD
of its rows, where each row is a local vector.
Since each row is represented by a local vector, the number of columns is
limited by the integer range but it should be much smaller in practice.</p>

<div class="codetabs">
<div data-lang="scala">

    <p>A <a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.RowMatrix"><code>RowMatrix</code></a> can be
created from an <code>RDD[Vector]</code> instance.  Then we can compute its column summary statistics and decompositions.
<a href="https://en.wikipedia.org/wiki/QR_decomposition">QR decomposition</a> is of the form A = QR where Q is an orthogonal matrix and R is an upper triangular matrix.
For <a href="https://en.wikipedia.org/wiki/Singular_value_decomposition">singular value decomposition (SVD)</a> and <a href="https://en.wikipedia.org/wiki/Principal_component_analysis">principal component analysis (PCA)</a>, please refer to <a href="mllib-dimensionality-reduction.html">Dimensionality reduction</a>.</p>

    <p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.RowMatrix"><code>RowMatrix</code> Scala docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.RowMatrix</span>

<span class="k">val</span> <span class="n">rows</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// an RDD of local vectors</span>
<span class="c1">// Create a RowMatrix from an RDD[Vector].</span>
<span class="k">val</span> <span class="n">mat</span><span class="k">:</span> <span class="kt">RowMatrix</span> <span class="o">=</span> <span class="k">new</span> <span class="nc">RowMatrix</span><span class="o">(</span><span class="n">rows</span><span class="o">)</span>

<span class="c1">// Get its size.</span>
<span class="k">val</span> <span class="n">m</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numRows</span><span class="o">()</span>
<span class="k">val</span> <span class="n">n</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numCols</span><span class="o">()</span>

<span class="c1">// QR decomposition </span>
<span class="k">val</span> <span class="n">qrResult</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">tallSkinnyQR</span><span class="o">(</span><span class="kc">true</span><span class="o">)</span></code></pre></div>

  </div>

<div data-lang="java">

    <p>A <a href="api/java/org/apache/spark/mllib/linalg/distributed/RowMatrix.html"><code>RowMatrix</code></a> can be
created from a <code>JavaRDD&lt;Vector&gt;</code> instance.  Then we can compute its column summary statistics.</p>

    <p>Refer to the <a href="api/java/org/apache/spark/mllib/linalg/distributed/RowMatrix.html"><code>RowMatrix</code> Java docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaRDD</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.RowMatrix</span><span class="o">;</span>

<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">Vector</span><span class="o">&gt;</span> <span class="n">rows</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a JavaRDD of local vectors</span>
<span class="c1">// Create a RowMatrix from an JavaRDD&lt;Vector&gt;.</span>
<span class="n">RowMatrix</span> <span class="n">mat</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">RowMatrix</span><span class="o">(</span><span class="n">rows</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span>

<span class="c1">// Get its size.</span>
<span class="kt">long</span> <span class="n">m</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">numRows</span><span class="o">();</span>
<span class="kt">long</span> <span class="n">n</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">numCols</span><span class="o">();</span>

<span class="c1">// QR decomposition </span>
<span class="n">QRDecomposition</span><span class="o">&lt;</span><span class="n">RowMatrix</span><span class="o">,</span> <span class="n">Matrix</span><span class="o">&gt;</span> <span class="n">result</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">tallSkinnyQR</span><span class="o">(</span><span class="kc">true</span><span class="o">);</span></code></pre></div>

  </div>

<div data-lang="python">

    <p>A <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.distributed.RowMatrix"><code>RowMatrix</code></a> can be 
created from an <code>RDD</code> of vectors.</p>

    <p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.distributed.RowMatrix"><code>RowMatrix</code> Python docs</a> for more details on the API.</p>

    <div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.linalg.distributed</span> <span class="kn">import</span> <span class="n">RowMatrix</span>

<span class="c"># Create an RDD of vectors.</span>
<span class="n">rows</span> <span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">parallelize</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">]])</span>

<span class="c"># Create a RowMatrix from an RDD of vectors.</span>
<span class="n">mat</span> <span class="o">=</span> <span class="n">RowMatrix</span><span class="p">(</span><span class="n">rows</span><span class="p">)</span>

<span class="c"># Get its size.</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numRows</span><span class="p">()</span>  <span class="c"># 4</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numCols</span><span class="p">()</span>  <span class="c"># 3</span>

<span class="c"># Get the rows as an RDD of vectors again.</span>
<span class="n">rowsRDD</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">rows</span></code></pre></div>

  </div>

</div>

<h3 id="indexedrowmatrix">IndexedRowMatrix</h3>

<p>An <code>IndexedRowMatrix</code> is similar to a <code>RowMatrix</code> but with meaningful row indices.  It is backed by
an RDD of indexed rows, so that each row is represented by its index (long-typed) and a local 
vector.</p>

<div class="codetabs">
<div data-lang="scala">

    <p>An
<a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.IndexedRowMatrix"><code>IndexedRowMatrix</code></a>
can be created from an <code>RDD[IndexedRow]</code> instance, where
<a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.IndexedRow"><code>IndexedRow</code></a> is a
wrapper over <code>(Long, Vector)</code>.  An <code>IndexedRowMatrix</code> can be converted to a <code>RowMatrix</code> by dropping
its row indices.</p>

    <p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.IndexedRowMatrix"><code>IndexedRowMatrix</code> Scala docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.</span><span class="o">{</span><span class="nc">IndexedRow</span><span class="o">,</span> <span class="nc">IndexedRowMatrix</span><span class="o">,</span> <span class="nc">RowMatrix</span><span class="o">}</span>

<span class="k">val</span> <span class="n">rows</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">IndexedRow</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// an RDD of indexed rows</span>
<span class="c1">// Create an IndexedRowMatrix from an RDD[IndexedRow].</span>
<span class="k">val</span> <span class="n">mat</span><span class="k">:</span> <span class="kt">IndexedRowMatrix</span> <span class="o">=</span> <span class="k">new</span> <span class="nc">IndexedRowMatrix</span><span class="o">(</span><span class="n">rows</span><span class="o">)</span>

<span class="c1">// Get its size.</span>
<span class="k">val</span> <span class="n">m</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numRows</span><span class="o">()</span>
<span class="k">val</span> <span class="n">n</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numCols</span><span class="o">()</span>

<span class="c1">// Drop its row indices.</span>
<span class="k">val</span> <span class="n">rowMat</span><span class="k">:</span> <span class="kt">RowMatrix</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toRowMatrix</span><span class="o">()</span></code></pre></div>

  </div>

<div data-lang="java">

    <p>An
<a href="api/java/org/apache/spark/mllib/linalg/distributed/IndexedRowMatrix.html"><code>IndexedRowMatrix</code></a>
can be created from an <code>JavaRDD&lt;IndexedRow&gt;</code> instance, where
<a href="api/java/org/apache/spark/mllib/linalg/distributed/IndexedRow.html"><code>IndexedRow</code></a> is a
wrapper over <code>(long, Vector)</code>.  An <code>IndexedRowMatrix</code> can be converted to a <code>RowMatrix</code> by dropping
its row indices.</p>

    <p>Refer to the <a href="api/java/org/apache/spark/mllib/linalg/distributed/IndexedRowMatrix.html"><code>IndexedRowMatrix</code> Java docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaRDD</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.IndexedRow</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.IndexedRowMatrix</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.RowMatrix</span><span class="o">;</span>

<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">IndexedRow</span><span class="o">&gt;</span> <span class="n">rows</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a JavaRDD of indexed rows</span>
<span class="c1">// Create an IndexedRowMatrix from a JavaRDD&lt;IndexedRow&gt;.</span>
<span class="n">IndexedRowMatrix</span> <span class="n">mat</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">IndexedRowMatrix</span><span class="o">(</span><span class="n">rows</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span>

<span class="c1">// Get its size.</span>
<span class="kt">long</span> <span class="n">m</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">numRows</span><span class="o">();</span>
<span class="kt">long</span> <span class="n">n</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">numCols</span><span class="o">();</span>

<span class="c1">// Drop its row indices.</span>
<span class="n">RowMatrix</span> <span class="n">rowMat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">toRowMatrix</span><span class="o">();</span></code></pre></div>

  </div>

<div data-lang="python">

    <p>An <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.distributed.IndexedRowMatrix"><code>IndexedRowMatrix</code></a>
can be created from an <code>RDD</code> of <code>IndexedRow</code>s, where 
<a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.distributed.IndexedRow"><code>IndexedRow</code></a> is a 
wrapper over <code>(long, vector)</code>.  An <code>IndexedRowMatrix</code> can be converted to a <code>RowMatrix</code> by dropping
its row indices.</p>

    <p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.distributed.IndexedRowMatrix"><code>IndexedRowMatrix</code> Python docs</a> for more details on the API.</p>

    <div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.linalg.distributed</span> <span class="kn">import</span> <span class="n">IndexedRow</span><span class="p">,</span> <span class="n">IndexedRowMatrix</span>

<span class="c"># Create an RDD of indexed rows.</span>
<span class="c">#   - This can be done explicitly with the IndexedRow class:</span>
<span class="n">indexedRows</span> <span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">parallelize</span><span class="p">([</span><span class="n">IndexedRow</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]),</span> 
                              <span class="n">IndexedRow</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]),</span> 
                              <span class="n">IndexedRow</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">]),</span> 
                              <span class="n">IndexedRow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">])])</span>
<span class="c">#   - or by using (long, vector) tuples:</span>
<span class="n">indexedRows</span> <span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">parallelize</span><span class="p">([(</span><span class="mi">0</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]),</span> 
                              <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">]),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">])])</span>

<span class="c"># Create an IndexedRowMatrix from an RDD of IndexedRows.</span>
<span class="n">mat</span> <span class="o">=</span> <span class="n">IndexedRowMatrix</span><span class="p">(</span><span class="n">indexedRows</span><span class="p">)</span>

<span class="c"># Get its size.</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numRows</span><span class="p">()</span>  <span class="c"># 4</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numCols</span><span class="p">()</span>  <span class="c"># 3</span>

<span class="c"># Get the rows as an RDD of IndexedRows.</span>
<span class="n">rowsRDD</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">rows</span>

<span class="c"># Convert to a RowMatrix by dropping the row indices.</span>
<span class="n">rowMat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toRowMatrix</span><span class="p">()</span>

<span class="c"># Convert to a CoordinateMatrix.</span>
<span class="n">coordinateMat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toCoordinateMatrix</span><span class="p">()</span>

<span class="c"># Convert to a BlockMatrix.</span>
<span class="n">blockMat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toBlockMatrix</span><span class="p">()</span></code></pre></div>

  </div>

</div>

<h3 id="coordinatematrix">CoordinateMatrix</h3>

<p>A <code>CoordinateMatrix</code> is a distributed matrix backed by an RDD of its entries.  Each entry is a tuple
of <code>(i: Long, j: Long, value: Double)</code>, where <code>i</code> is the row index, <code>j</code> is the column index, and
<code>value</code> is the entry value.  A <code>CoordinateMatrix</code> should be used only when both
dimensions of the matrix are huge and the matrix is very sparse.</p>

<div class="codetabs">
<div data-lang="scala">

    <p>A
<a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.CoordinateMatrix"><code>CoordinateMatrix</code></a>
can be created from an <code>RDD[MatrixEntry]</code> instance, where
<a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.MatrixEntry"><code>MatrixEntry</code></a> is a
wrapper over <code>(Long, Long, Double)</code>.  A <code>CoordinateMatrix</code> can be converted to an <code>IndexedRowMatrix</code>
with sparse rows by calling <code>toIndexedRowMatrix</code>.  Other computations for 
<code>CoordinateMatrix</code> are not currently supported.</p>

    <p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.CoordinateMatrix"><code>CoordinateMatrix</code> Scala docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.</span><span class="o">{</span><span class="nc">CoordinateMatrix</span><span class="o">,</span> <span class="nc">MatrixEntry</span><span class="o">}</span>

<span class="k">val</span> <span class="n">entries</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">MatrixEntry</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// an RDD of matrix entries</span>
<span class="c1">// Create a CoordinateMatrix from an RDD[MatrixEntry].</span>
<span class="k">val</span> <span class="n">mat</span><span class="k">:</span> <span class="kt">CoordinateMatrix</span> <span class="o">=</span> <span class="k">new</span> <span class="nc">CoordinateMatrix</span><span class="o">(</span><span class="n">entries</span><span class="o">)</span>

<span class="c1">// Get its size.</span>
<span class="k">val</span> <span class="n">m</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numRows</span><span class="o">()</span>
<span class="k">val</span> <span class="n">n</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numCols</span><span class="o">()</span>

<span class="c1">// Convert it to an IndexRowMatrix whose rows are sparse vectors.</span>
<span class="k">val</span> <span class="n">indexedRowMatrix</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toIndexedRowMatrix</span><span class="o">()</span></code></pre></div>

  </div>

<div data-lang="java">

    <p>A
<a href="api/java/org/apache/spark/mllib/linalg/distributed/CoordinateMatrix.html"><code>CoordinateMatrix</code></a>
can be created from a <code>JavaRDD&lt;MatrixEntry&gt;</code> instance, where
<a href="api/java/org/apache/spark/mllib/linalg/distributed/MatrixEntry.html"><code>MatrixEntry</code></a> is a
wrapper over <code>(long, long, double)</code>.  A <code>CoordinateMatrix</code> can be converted to an <code>IndexedRowMatrix</code>
with sparse rows by calling <code>toIndexedRowMatrix</code>. Other computations for 
<code>CoordinateMatrix</code> are not currently supported.</p>

    <p>Refer to the <a href="api/java/org/apache/spark/mllib/linalg/distributed/CoordinateMatrix.html"><code>CoordinateMatrix</code> Java docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaRDD</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.CoordinateMatrix</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.IndexedRowMatrix</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.MatrixEntry</span><span class="o">;</span>

<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">MatrixEntry</span><span class="o">&gt;</span> <span class="n">entries</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a JavaRDD of matrix entries</span>
<span class="c1">// Create a CoordinateMatrix from a JavaRDD&lt;MatrixEntry&gt;.</span>
<span class="n">CoordinateMatrix</span> <span class="n">mat</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">CoordinateMatrix</span><span class="o">(</span><span class="n">entries</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span>

<span class="c1">// Get its size.</span>
<span class="kt">long</span> <span class="n">m</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">numRows</span><span class="o">();</span>
<span class="kt">long</span> <span class="n">n</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">numCols</span><span class="o">();</span>

<span class="c1">// Convert it to an IndexRowMatrix whose rows are sparse vectors.</span>
<span class="n">IndexedRowMatrix</span> <span class="n">indexedRowMatrix</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">toIndexedRowMatrix</span><span class="o">();</span></code></pre></div>

  </div>

<div data-lang="python">

    <p>A <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.distributed.CoordinateMatrix"><code>CoordinateMatrix</code></a>
can be created from an <code>RDD</code> of <code>MatrixEntry</code> entries, where 
<a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.distributed.MatrixEntry"><code>MatrixEntry</code></a> is a 
wrapper over <code>(long, long, float)</code>.  A <code>CoordinateMatrix</code> can be converted to a <code>RowMatrix</code> by 
calling <code>toRowMatrix</code>, or to an <code>IndexedRowMatrix</code> with sparse rows by calling <code>toIndexedRowMatrix</code>.</p>

    <p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.distributed.CoordinateMatrix"><code>CoordinateMatrix</code> Python docs</a> for more details on the API.</p>

    <div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.linalg.distributed</span> <span class="kn">import</span> <span class="n">CoordinateMatrix</span><span class="p">,</span> <span class="n">MatrixEntry</span>

<span class="c"># Create an RDD of coordinate entries.</span>
<span class="c">#   - This can be done explicitly with the MatrixEntry class:</span>
<span class="n">entries</span> <span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">parallelize</span><span class="p">([</span><span class="n">MatrixEntry</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">),</span> <span class="n">MatrixEntry</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">),</span> <span class="n">MatrixEntry</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">3.7</span><span class="p">)])</span>
<span class="c">#   - or using (long, long, float) tuples:</span>
<span class="n">entries</span> <span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">parallelize</span><span class="p">([(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">3.7</span><span class="p">)])</span>

<span class="c"># Create an CoordinateMatrix from an RDD of MatrixEntries.</span>
<span class="n">mat</span> <span class="o">=</span> <span class="n">CoordinateMatrix</span><span class="p">(</span><span class="n">entries</span><span class="p">)</span>

<span class="c"># Get its size.</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numRows</span><span class="p">()</span>  <span class="c"># 3</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numCols</span><span class="p">()</span>  <span class="c"># 2</span>

<span class="c"># Get the entries as an RDD of MatrixEntries.</span>
<span class="n">entriesRDD</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">entries</span>

<span class="c"># Convert to a RowMatrix.</span>
<span class="n">rowMat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toRowMatrix</span><span class="p">()</span>

<span class="c"># Convert to an IndexedRowMatrix.</span>
<span class="n">indexedRowMat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toIndexedRowMatrix</span><span class="p">()</span>

<span class="c"># Convert to a BlockMatrix.</span>
<span class="n">blockMat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toBlockMatrix</span><span class="p">()</span></code></pre></div>

  </div>

</div>

<h3 id="blockmatrix">BlockMatrix</h3>

<p>A <code>BlockMatrix</code> is a distributed matrix backed by an RDD of <code>MatrixBlock</code>s, where a <code>MatrixBlock</code> is
a tuple of <code>((Int, Int), Matrix)</code>, where the <code>(Int, Int)</code> is the index of the block, and <code>Matrix</code> is
the sub-matrix at the given index with size <code>rowsPerBlock</code> x <code>colsPerBlock</code>.
<code>BlockMatrix</code> supports methods such as <code>add</code> and <code>multiply</code> with another <code>BlockMatrix</code>.
<code>BlockMatrix</code> also has a helper function <code>validate</code> which can be used to check whether the
<code>BlockMatrix</code> is set up properly.</p>

<div class="codetabs">
<div data-lang="scala">

    <p>A <a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.BlockMatrix"><code>BlockMatrix</code></a> can be
most easily created from an <code>IndexedRowMatrix</code> or <code>CoordinateMatrix</code> by calling <code>toBlockMatrix</code>.
<code>toBlockMatrix</code> creates blocks of size 1024 x 1024 by default.
Users may change the block size by supplying the values through <code>toBlockMatrix(rowsPerBlock, colsPerBlock)</code>.</p>

    <p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.BlockMatrix"><code>BlockMatrix</code> Scala docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.</span><span class="o">{</span><span class="nc">BlockMatrix</span><span class="o">,</span> <span class="nc">CoordinateMatrix</span><span class="o">,</span> <span class="nc">MatrixEntry</span><span class="o">}</span>

<span class="k">val</span> <span class="n">entries</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">MatrixEntry</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// an RDD of (i, j, v) matrix entries</span>
<span class="c1">// Create a CoordinateMatrix from an RDD[MatrixEntry].</span>
<span class="k">val</span> <span class="n">coordMat</span><span class="k">:</span> <span class="kt">CoordinateMatrix</span> <span class="o">=</span> <span class="k">new</span> <span class="nc">CoordinateMatrix</span><span class="o">(</span><span class="n">entries</span><span class="o">)</span>
<span class="c1">// Transform the CoordinateMatrix to a BlockMatrix</span>
<span class="k">val</span> <span class="n">matA</span><span class="k">:</span> <span class="kt">BlockMatrix</span> <span class="o">=</span> <span class="n">coordMat</span><span class="o">.</span><span class="n">toBlockMatrix</span><span class="o">().</span><span class="n">cache</span><span class="o">()</span>

<span class="c1">// Validate whether the BlockMatrix is set up properly. Throws an Exception when it is not valid.</span>
<span class="c1">// Nothing happens if it is valid.</span>
<span class="n">matA</span><span class="o">.</span><span class="n">validate</span><span class="o">()</span>

<span class="c1">// Calculate A^T A.</span>
<span class="k">val</span> <span class="n">ata</span> <span class="k">=</span> <span class="n">matA</span><span class="o">.</span><span class="n">transpose</span><span class="o">.</span><span class="n">multiply</span><span class="o">(</span><span class="n">matA</span><span class="o">)</span></code></pre></div>

  </div>

<div data-lang="java">

    <p>A <a href="api/java/org/apache/spark/mllib/linalg/distributed/BlockMatrix.html"><code>BlockMatrix</code></a> can be
most easily created from an <code>IndexedRowMatrix</code> or <code>CoordinateMatrix</code> by calling <code>toBlockMatrix</code>.
<code>toBlockMatrix</code> creates blocks of size 1024 x 1024 by default.
Users may change the block size by supplying the values through <code>toBlockMatrix(rowsPerBlock, colsPerBlock)</code>.</p>

    <p>Refer to the <a href="api/java/org/apache/spark/mllib/linalg/distributed/BlockMatrix.html"><code>BlockMatrix</code> Java docs</a> for details on the API.</p>

    <div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaRDD</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.BlockMatrix</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.CoordinateMatrix</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.IndexedRowMatrix</span><span class="o">;</span>

<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">MatrixEntry</span><span class="o">&gt;</span> <span class="n">entries</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a JavaRDD of (i, j, v) Matrix Entries</span>
<span class="c1">// Create a CoordinateMatrix from a JavaRDD&lt;MatrixEntry&gt;.</span>
<span class="n">CoordinateMatrix</span> <span class="n">coordMat</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">CoordinateMatrix</span><span class="o">(</span><span class="n">entries</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span>
<span class="c1">// Transform the CoordinateMatrix to a BlockMatrix</span>
<span class="n">BlockMatrix</span> <span class="n">matA</span> <span class="o">=</span> <span class="n">coordMat</span><span class="o">.</span><span class="na">toBlockMatrix</span><span class="o">().</span><span class="na">cache</span><span class="o">();</span>

<span class="c1">// Validate whether the BlockMatrix is set up properly. Throws an Exception when it is not valid.</span>
<span class="c1">// Nothing happens if it is valid.</span>
<span class="n">matA</span><span class="o">.</span><span class="na">validate</span><span class="o">();</span>

<span class="c1">// Calculate A^T A.</span>
<span class="n">BlockMatrix</span> <span class="n">ata</span> <span class="o">=</span> <span class="n">matA</span><span class="o">.</span><span class="na">transpose</span><span class="o">().</span><span class="na">multiply</span><span class="o">(</span><span class="n">matA</span><span class="o">);</span></code></pre></div>

  </div>

<div data-lang="python">

    <p>A <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.distributed.BlockMatrix"><code>BlockMatrix</code></a> 
can be created from an <code>RDD</code> of sub-matrix blocks, where a sub-matrix block is a 
<code>((blockRowIndex, blockColIndex), sub-matrix)</code> tuple.</p>

    <p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.linalg.distributed.BlockMatrix"><code>BlockMatrix</code> Python docs</a> for more details on the API.</p>

    <div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.linalg</span> <span class="kn">import</span> <span class="n">Matrices</span>
<span class="kn">from</span> <span class="nn">pyspark.mllib.linalg.distributed</span> <span class="kn">import</span> <span class="n">BlockMatrix</span>

<span class="c"># Create an RDD of sub-matrix blocks.</span>
<span class="n">blocks</span> <span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">parallelize</span><span class="p">([((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Matrices</span><span class="o">.</span><span class="n">dense</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">])),</span> 
                         <span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Matrices</span><span class="o">.</span><span class="n">dense</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">]))])</span>

<span class="c"># Create a BlockMatrix from an RDD of sub-matrix blocks.</span>
<span class="n">mat</span> <span class="o">=</span> <span class="n">BlockMatrix</span><span class="p">(</span><span class="n">blocks</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>

<span class="c"># Get its size.</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numRows</span><span class="p">()</span> <span class="c"># 6</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numCols</span><span class="p">()</span> <span class="c"># 2</span>

<span class="c"># Get the blocks as an RDD of sub-matrix blocks.</span>
<span class="n">blocksRDD</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">blocks</span>

<span class="c"># Convert to a LocalMatrix.</span>
<span class="n">localMat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toLocalMatrix</span><span class="p">()</span>

<span class="c"># Convert to an IndexedRowMatrix.</span>
<span class="n">indexedRowMat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toIndexedRowMatrix</span><span class="p">()</span>

<span class="c"># Convert to a CoordinateMatrix.</span>
<span class="n">coordinateMat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toCoordinateMatrix</span><span class="p">()</span></code></pre></div>

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